Terms on both sides dyer and hinterbichler 2009

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Unformatted text preview: n, H ¼ h ; 10 ðg À r N À r N Þ; 2N  (9.17) where a prime means a derivative with respect to y. The action is now16  Z M3 Z pffiffiffiffiffiffiffi ffi S¼ 5 þ d4 xdyN Àg½RðgÞ þ K 2 À K K  Š 2 L R Z 4 þ d xL4 : (9.20) The Ricci scalar and metric determinant are ð4Þ ð5ÞR ¼ R þ ðK2 À K K  Þ þ 2rA ðnB rB nA À nA K Þ; (9.18) pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ÀG ¼ N Àg: (9.19) The total derivatives coming from 2rA ðn rB n À n K Þ in the Einstein-Hilbert part of the action exactly cancel the GibbonsHawking terms. Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 H55 ¼ 2n: (9.23) À2RAB ðGÞlinear ¼ hð5Þ HAB þ @A @B H À @C @A HBC À @C @B HAC ¼ 0: (9.24) We solve Eq. (9.24) in the de Donder gauge, @B HAB À [email protected] H ¼ 0: 2 (9.25) With this, Eq. (9.24) is equivalent to hð5Þ HAB ¼ 0; (9.26) along with the de Donder gauge condition (9.25). In terms of the ADM variables, Eq. (9.26) becomes A A hh þ @2 h ¼ 0; y (9.27) hn þ @2 n ¼ 0; y (9.28) hn þ @2 n ¼ 0; y It can be checked that a flat brane living in flat space is a solution to the equations of motion of this action. This is called the normal branch. There is another maximally B H5 ¼ n ; We first expand the DGP action (9.20) to quadratic order in h , n , and n. We then solve the 5d equations of motion, subject to arbitrary boundary values on the brane and going to zero at infinity. We then plug this solution back into the action to obtain an effective 4d theory for the arbitrary brane boundary values. The 5d equations of motion away from the brane are simply the vacuum Einstein equations, which read, to linear order, The 4d extrinsic curvature is taken with respect to the positive pointing normal nL and is given by 16 N ¼ n ; R where KR and KL are the extrinsic curvatures relative to the normals nR and nL , respectively. We now go to spacelike ADM variables (Arnowitt, Deser, and Misner, 1960, 1962) adapted to the brane [see Poisson (2004) and Dyer and Hinterbichler (2009) for detailed derivations and formulas]. The lapse and shift relative to y are N  ðx; yÞ and N ðx; yÞ, and the 4d metric is g ðx; yÞ. The 5d metric is ! N 2 þ N  N N GAB ¼ : (9.16) N g K ¼ (9.21) (9.29) where h is the 4d Laplacian. These have the following solutions in terms of boundary values h ðxÞ, n ðxÞ, and nðxÞ: h ðx; yÞ ¼ eÀyÁ h ðxÞ; (9.30) n ðx; yÞ ¼ eÀyÁ n ðxÞ; (9.31) Kurt Hinterbichler: Theoretical aspects of massive gravity 704 nðx; yÞ ¼ eÀyÁ nðxÞ: (9.32) Here the operator Á is the formal square root of the 4d Laplacian, pffiffiffiffiffiffiffiffiffi Á  Àh: (9.33) The A ¼  and A ¼ 5 components of the gauge condition (9.25) are, respectively, @ h À 1@ h þ @y n À @ n ¼ 0; 2 (9.34) @ n À (9.35) 1 [email protected] h þ @ y n ¼ 0: For these to be satisfied everywhere, it is necessary and sufficient that the boundary fields satisfy the following at y ¼ 0: @ h À 1@ h ...
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